# E8 — the vertical claim: n-parent recombination exceeds any parent (Fisher–Muller)

**Claim tested.** The society's headline, and the thing no fixed-`p*` model could express: can an
offspring recombined from **many decorrelated parents** be *fitter than any parent* — capability that
**exceeds** every component, not merely recovers a ceiling?

**Setup.** A capability is a **genotype** of `L=12` biallelic loci; fitness = number of correct
loci; the optimum (all-correct, fitness 12) is a genotype **no parent possesses**. Each *parent* is a
specialist: confident-correct (`hi=0.9`) on the loci it has mastered, agnostic (`lo=0.45`) on the
rest. Which loci each masters comes from the exact shared-switch construction, so **parent count
`K_T`** and **decorrelation `ρ`** are clean, independently-swept knobs. Deployed capability = fitness
of the **mode** (most-probable) genotype — what you would ship. 40 replicates.

### Symbols
- **parent** = a specialist model; **offspring** = the recombined model; **`K_T`** = number of parents (unbounded — biological sex is stuck at 2; model merging is not).
- **`ρ`** = correlation of which loci parents master (0 = complementary, 1 = identical clones).
- **best parent** = fittest single specialist · **average** = mean-mixture "model soup" (combine, don't recombine) · **sexual** = union-preserving recombination (assemble the best allele of each locus).

### The two panels
1. **Recombination exceeds any parent (ρ=0).** Deployed capability vs `K_T`: **sexual (red) climbs
   to the optimum (12)** as parents accumulate — a genotype none of them had — while the **best single
   parent (grey) plateaus at ~8.7** and the **model soup (blue) reaches ~11.6** but is beaten by
   sexual at every `K_T` (and badly at small `K_T`: at 2 parents, sexual 9.0 vs soup 7.2 vs best 6.9).
2. **Decorrelation is the fuel.** Sexual capability vs `K_T` for `ρ ∈ {0, 0.5, 1}`: decorrelated
   parents (`ρ=0`) climb to the optimum; identical clones (`ρ=1`) buy nothing (flat at ~6). The
   benefit is *combinatorial reach across complementary parents*, not merely "more models".

### Takeaway
This is the Fisher–Muller effect for AI: recombination assembles beneficial variants that live in
*different* parents into an offspring fitter than any of them. It is the rigorous, un-preempted core
of the Lamarckian society — collapse is asexual degradation; the cure is **sex, with no parent
limit.** It reframes model merging from "averaging weights" to "meiotic reassortment", and it is the
mechanism by which general capability can *climb while each specialty is re-earned and exceeded.* The
dynamic version (why a lone lineage cannot do this) is E7. **Falsifier (not triggered):** if sexual
never exceeded the best parent, or averaging matched it, the vertical claim would fail.